Tropical independence I: Shapes of divisors and a proof of the Gieseker–Petri theorem

We develop a framework to apply tropical and nonarchimedean analytic methods to multiplication maps for linear series on algebraic curves, studying degenerations of these multiplications maps when the special fiber is not of compact type. As an application, we give a new proof of the Gieseker–Petri theorem, including an explicit tropical criterion for a curve over a valued field to be Gieseker–Petri general.